Question: Simplify the following expression and state the condition under which the simplification is valid: $t = \dfrac{q^2 - 8q}{q^2 - 3q - 40}$
Explanation: First factor the expressions in the numerator and denominator. $ \dfrac{q^2 - 8q}{q^2 - 3q - 40} = \dfrac{(q)(q - 8)}{(q + 5)(q - 8)} $ Notice that the term $(q - 8)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(q - 8)$ gives: $t = \dfrac{q}{q + 5}$ Since we divided by $(q - 8)$, $q \neq 8$. $t = \dfrac{q}{q + 5}; \space q \neq 8$